Automorphism Group of C Over R

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Theorem

The field extension $\C / \R$ of complex numbers $\C$ over real numbers $\R$ has automorphism group $\operatorname{Aut}$:

$\operatorname{Aut} \paren {\C / \R} = \set {\operatorname{id}, \sigma}$

where:

$\operatorname{id}$ denotes the identity mapping
$\sigma$ denotes complex conjugation


Proof